The main difficulty here is that when the vector fields \(V^i\)  are just Sobolev, the respective Lagrangian flows have only very weak regularity properties: in particular, they are just summable but not Sobolev (not even of fractional order)
as shown in [1]}, [2]}, so that their weak derivatives do not have any pointwise (even a.e.) meaning. This is in sharp contrast with the situation when \(V^i\)  are Lipschitz, since in the latter case it is a textbook result that the respective flows are Lipschitz too. That is why the technique we adopt in this paper is essentially different from that used in [3]}. In fact, in the latter paper a set-valued Lie bracket of Lipschitz vector fields is introduced and then it is shown, substantially, that the usual expansions (of course adapted now to the set-valued setting) of compositions of flow maps in each point are still valid for this case with the set-valued Lie bracket instead of the classical one.
As a side remark, we mention that another quite natural extension of both the classical Lie bracket and its set-valued version,
the measure valued Lie bracket, has been introduced recently in [4]}.
In the general case of Sobolev \(V^i\) , in view of the mentioned lack of regularity of flows, no technique based on pointwise
expansions would work and therefore we are forced to develop a completely different PDE/measure theory-style approach.
